statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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upper_semicontinuous_within_at_cinfi {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝[s] x, bdd_below (range $ λ i, f i y))
(h : ∀ i, upper_semicontinuous_within_at (f i) s x) :
upper_semicontinuous_within_at (λ x', ⨅ i, f i x') s x | @lower_semicontinuous_within_at_csupr α _ x s ι δ'ᵒᵈ _ f bdd h | lemma | upper_semicontinuous_within_at_cinfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_below",
"lower_semicontinuous_within_at_csupr",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_within_at_infi {f : ι → α → δ}
(h : ∀ i, upper_semicontinuous_within_at (f i) s x) :
upper_semicontinuous_within_at (λ x', ⨅ i, f i x') s x | @lower_semicontinuous_within_at_supr α _ x s ι δᵒᵈ _ f h | lemma | upper_semicontinuous_within_at_infi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_within_at_supr",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_within_at_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, upper_semicontinuous_within_at (f i hi) s x) :
upper_semicontinuous_within_at (λ x', ⨅ i hi, f i hi x') s x | upper_semicontinuous_within_at_infi $ λ i, upper_semicontinuous_within_at_infi $ λ hi, h i hi | lemma | upper_semicontinuous_within_at_binfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_within_at",
"upper_semicontinuous_within_at_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_at_cinfi {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝 x, bdd_below (range $ λ i, f i y))
(h : ∀ i, upper_semicontinuous_at (f i) x) :
upper_semicontinuous_at (λ x', ⨅ i, f i x') x | @lower_semicontinuous_at_csupr α _ x ι δ'ᵒᵈ _ f bdd h | lemma | upper_semicontinuous_at_cinfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_below",
"lower_semicontinuous_at_csupr",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_at_infi {f : ι → α → δ}
(h : ∀ i, upper_semicontinuous_at (f i) x) :
upper_semicontinuous_at (λ x', ⨅ i, f i x') x | @lower_semicontinuous_at_supr α _ x ι δᵒᵈ _ f h | lemma | upper_semicontinuous_at_infi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_at_supr",
"upper_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_at_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, upper_semicontinuous_at (f i hi) x) :
upper_semicontinuous_at (λ x', ⨅ i hi, f i hi x') x | upper_semicontinuous_at_infi $ λ i, upper_semicontinuous_at_infi $ λ hi, h i hi | lemma | upper_semicontinuous_at_binfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_at",
"upper_semicontinuous_at_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on_cinfi {f : ι → α → δ'}
(bdd : ∀ x ∈ s, bdd_below (range $ λ i, f i x))
(h : ∀ i, upper_semicontinuous_on (f i) s) :
upper_semicontinuous_on (λ x', ⨅ i, f i x') s | λ x hx, upper_semicontinuous_within_at_cinfi (eventually_nhds_within_of_forall bdd) (λ i, h i x hx) | lemma | upper_semicontinuous_on_cinfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_below",
"eventually_nhds_within_of_forall",
"upper_semicontinuous_on",
"upper_semicontinuous_within_at_cinfi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on_infi {f : ι → α → δ}
(h : ∀ i, upper_semicontinuous_on (f i) s) :
upper_semicontinuous_on (λ x', ⨅ i, f i x') s | λ x hx, upper_semicontinuous_within_at_infi (λ i, h i x hx) | lemma | upper_semicontinuous_on_infi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_on",
"upper_semicontinuous_within_at_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_on_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, upper_semicontinuous_on (f i hi) s) :
upper_semicontinuous_on (λ x', ⨅ i hi, f i hi x') s | upper_semicontinuous_on_infi $ λ i, upper_semicontinuous_on_infi $ λ hi, h i hi | lemma | upper_semicontinuous_on_binfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_on",
"upper_semicontinuous_on_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_cinfi {f : ι → α → δ'}
(bdd : ∀ x, bdd_below (range $ λ i, f i x))
(h : ∀ i, upper_semicontinuous (f i)) :
upper_semicontinuous (λ x', ⨅ i, f i x') | λ x, upper_semicontinuous_at_cinfi (eventually_of_forall bdd) (λ i, h i x) | lemma | upper_semicontinuous_cinfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"bdd_below",
"upper_semicontinuous",
"upper_semicontinuous_at_cinfi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_infi {f : ι → α → δ}
(h : ∀ i, upper_semicontinuous (f i)) :
upper_semicontinuous (λ x', ⨅ i, f i x') | λ x, upper_semicontinuous_at_infi (λ i, h i x) | lemma | upper_semicontinuous_infi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous",
"upper_semicontinuous_at_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_semicontinuous_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ}
(h : ∀ i hi, upper_semicontinuous (f i hi)) :
upper_semicontinuous (λ x', ⨅ i hi, f i hi x') | upper_semicontinuous_infi $ λ i, upper_semicontinuous_infi $ λ hi, h i hi | lemma | upper_semicontinuous_binfi | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous",
"upper_semicontinuous_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff_lower_upper_semicontinuous_within_at {f : α → γ} :
continuous_within_at f s x ↔
lower_semicontinuous_within_at f s x ∧ upper_semicontinuous_within_at f s x | begin
refine ⟨λ h, ⟨h.lower_semicontinuous_within_at, h.upper_semicontinuous_within_at⟩, _⟩,
rintros ⟨h₁, h₂⟩,
assume v hv,
simp only [filter.mem_map],
by_cases Hl : ∃ l, l < f x,
{ rcases exists_Ioc_subset_of_mem_nhds hv Hl with ⟨l, lfx, hl⟩,
by_cases Hu : ∃ u, f x < u,
{ rcases exists_Ico_subset_o... | lemma | continuous_within_at_iff_lower_upper_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_within_at",
"exists_Ico_subset_of_mem_nhds",
"exists_Ioc_subset_of_mem_nhds",
"filter.eventually_of_forall",
"filter.mem_map",
"lower_semicontinuous_within_at",
"mem_of_mem_nhds",
"not_exists",
"upper_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff_lower_upper_semicontinuous_at {f : α → γ} :
continuous_at f x ↔ (lower_semicontinuous_at f x ∧ upper_semicontinuous_at f x) | by simp_rw [← continuous_within_at_univ, ← lower_semicontinuous_within_at_univ_iff,
← upper_semicontinuous_within_at_univ_iff,
continuous_within_at_iff_lower_upper_semicontinuous_within_at] | lemma | continuous_at_iff_lower_upper_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_at",
"continuous_within_at_iff_lower_upper_semicontinuous_within_at",
"continuous_within_at_univ",
"lower_semicontinuous_at",
"lower_semicontinuous_within_at_univ_iff",
"upper_semicontinuous_at",
"upper_semicontinuous_within_at_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff_lower_upper_semicontinuous_on {f : α → γ} :
continuous_on f s ↔ (lower_semicontinuous_on f s ∧ upper_semicontinuous_on f s) | begin
simp only [continuous_on, continuous_within_at_iff_lower_upper_semicontinuous_within_at],
exact ⟨λ H, ⟨λ x hx, (H x hx).1, λ x hx, (H x hx).2⟩, λ H x hx, ⟨H.1 x hx, H.2 x hx⟩⟩
end | lemma | continuous_on_iff_lower_upper_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous_on",
"continuous_within_at_iff_lower_upper_semicontinuous_within_at",
"lower_semicontinuous_on",
"upper_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_lower_upper_semicontinuous {f : α → γ} :
continuous f ↔ (lower_semicontinuous f ∧ upper_semicontinuous f) | by simp_rw [continuous_iff_continuous_on_univ, continuous_on_iff_lower_upper_semicontinuous_on,
lower_semicontinuous_on_univ_iff, upper_semicontinuous_on_univ_iff] | lemma | continuous_iff_lower_upper_semicontinuous | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on_iff_lower_upper_semicontinuous_on",
"lower_semicontinuous",
"lower_semicontinuous_on_univ_iff",
"upper_semicontinuous",
"upper_semicontinuous_on_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separated_nhds : set α → set α → Prop | λ (s t : set α), ∃ U V : (set α), (is_open U) ∧ is_open V ∧
(s ⊆ U) ∧ (t ⊆ V) ∧ disjoint U V | def | separated_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"is_open"
] | `separated_nhds` is a predicate on pairs of sub`set`s of a topological space. It holds if the two
sub`set`s are contained in disjoint open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_nhds_iff_disjoint {s t : set α} :
separated_nhds s t ↔ disjoint (𝓝ˢ s) (𝓝ˢ t) | by simp only [(has_basis_nhds_set s).disjoint_iff (has_basis_nhds_set t), separated_nhds,
exists_prop, ← exists_and_distrib_left, and.assoc, and.comm, and.left_comm] | lemma | separated_nhds_iff_disjoint | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"disjoint_iff",
"exists_and_distrib_left",
"exists_prop",
"has_basis_nhds_set",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm : separated_nhds s t → separated_nhds t s | λ ⟨U, V, oU, oV, aU, bV, UV⟩, ⟨V, U, oV, oU, bV, aU, disjoint.symm UV⟩ | lemma | separated_nhds.symm | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint.symm",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm (s t : set α) : separated_nhds s t ↔ separated_nhds t s | ⟨symm, symm⟩ | lemma | separated_nhds.comm | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"comm",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage [topological_space β] {f : α → β} {s t : set β} (h : separated_nhds s t)
(hf : continuous f) : separated_nhds (f ⁻¹' s) (f ⁻¹' t) | let ⟨U, V, oU, oV, sU, tV, UV⟩ := h in
⟨f ⁻¹' U, f ⁻¹' V, oU.preimage hf, oV.preimage hf, preimage_mono sU, preimage_mono tV,
UV.preimage f⟩ | lemma | separated_nhds.preimage | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"separated_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint (h : separated_nhds s t) : disjoint s t | let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h in hd.mono hsU htV | lemma | separated_nhds.disjoint | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_closure_left (h : separated_nhds s t) : disjoint (closure s) t | let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h
in (hd.closure_left hV).mono (closure_mono hsU) htV | lemma | separated_nhds.disjoint_closure_left | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"closure_mono",
"disjoint",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_closure_right (h : separated_nhds s t) : disjoint s (closure t) | h.symm.disjoint_closure_left.symm | lemma | separated_nhds.disjoint_closure_right | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_right (s : set α) : separated_nhds s ∅ | ⟨_, _, is_open_univ, is_open_empty, λ a h, mem_univ a, λ a h, by cases h, disjoint_empty _⟩ | lemma | separated_nhds.empty_right | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open_empty",
"is_open_univ",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_left (s : set α) : separated_nhds ∅ s | (empty_right _).symm | lemma | separated_nhds.empty_left | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono (h : separated_nhds s₂ t₂) (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : separated_nhds s₁ t₁ | let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h in ⟨U, V, hU, hV, hs.trans hsU, ht.trans htV, hd⟩ | lemma | separated_nhds.mono | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_left : separated_nhds s u → separated_nhds t u → separated_nhds (s ∪ t) u | by simpa only [separated_nhds_iff_disjoint, nhds_set_union, disjoint_sup_left] using and.intro | lemma | separated_nhds.union_left | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint_sup_left",
"nhds_set_union",
"separated_nhds",
"separated_nhds_iff_disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_right (ht : separated_nhds s t) (hu : separated_nhds s u) :
separated_nhds s (t ∪ u) | (ht.symm.union_left hu.symm).symm | lemma | separated_nhds.union_right | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space (α : Type u) [topological_space α] : Prop | (t0 : ∀ ⦃x y : α⦄, inseparable x y → x = y) | class | t0_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable",
"topological_space"
] | A T₀ space, also known as a Kolmogorov space, is a topological space such that for every pair
`x ≠ y`, there is an open set containing one but not the other. We formulate the definition in terms
of the `inseparable` relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t0_space_iff_inseparable (α : Type u) [topological_space α] :
t0_space α ↔ ∀ (x y : α), inseparable x y → x = y | ⟨λ ⟨h⟩, h, λ h, ⟨h⟩⟩ | lemma | t0_space_iff_inseparable | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable",
"t0_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space_iff_not_inseparable (α : Type u) [topological_space α] :
t0_space α ↔ ∀ (x y : α), x ≠ y → ¬inseparable x y | by simp only [t0_space_iff_inseparable, ne.def, not_imp_not] | lemma | t0_space_iff_not_inseparable | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable",
"not_imp_not",
"t0_space",
"t0_space_iff_inseparable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable.eq [t0_space α] {x y : α} (h : inseparable x y) : x = y | t0_space.t0 h | lemma | inseparable.eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.injective [topological_space β] [t0_space α] {f : α → β}
(hf : inducing f) : injective f | λ x y h, inseparable.eq $ hf.inseparable_iff.1 $ h ▸ inseparable.refl _ | lemma | inducing.injective | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inducing",
"inseparable.eq",
"inseparable.refl",
"t0_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.embedding [topological_space β] [t0_space α] {f : α → β}
(hf : inducing f) : embedding f | ⟨hf, hf.injective⟩ | lemma | inducing.embedding | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"embedding",
"inducing",
"t0_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_iff_inducing [topological_space β] [t0_space α] {f : α → β} :
embedding f ↔ inducing f | ⟨embedding.to_inducing, inducing.embedding⟩ | lemma | embedding_iff_inducing | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"embedding",
"inducing",
"t0_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space_iff_nhds_injective (α : Type u) [topological_space α] :
t0_space α ↔ injective (𝓝 : α → filter α) | t0_space_iff_inseparable α | lemma | t0_space_iff_nhds_injective | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"t0_space",
"t0_space_iff_inseparable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_injective [t0_space α] : injective (𝓝 : α → filter α) | (t0_space_iff_nhds_injective α).1 ‹_› | lemma | nhds_injective | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"t0_space",
"t0_space_iff_nhds_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff_eq [t0_space α] {x y : α} : inseparable x y ↔ x = y | nhds_injective.eq_iff | lemma | inseparable_iff_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_nhds_iff [t0_space α] {a b : α} : 𝓝 a = 𝓝 b ↔ a = b | nhds_injective.eq_iff | lemma | nhds_eq_nhds_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_eq_eq [t0_space α] : inseparable = @eq α | funext₂ $ λ x y, propext inseparable_iff_eq | lemma | inseparable_eq_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"funext₂",
"inseparable",
"inseparable_iff_eq",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space_iff_exists_is_open_xor_mem (α : Type u) [topological_space α] :
t0_space α ↔ ∀ x y, x ≠ y → ∃ U:set α, is_open U ∧ (xor (x ∈ U) (y ∈ U)) | by simp only [t0_space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
inseparable_iff_forall_open] | lemma | t0_space_iff_exists_is_open_xor_mem | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"exists_prop",
"inseparable_iff_forall_open",
"is_open",
"not_forall",
"t0_space",
"t0_space_iff_not_inseparable",
"topological_space",
"xor_iff_not_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_is_open_xor_mem [t0_space α] {x y : α} (h : x ≠ y) :
∃ U : set α, is_open U ∧ xor (x ∈ U) (y ∈ U) | (t0_space_iff_exists_is_open_xor_mem α).1 ‹_› x y h | lemma | exists_is_open_xor_mem | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"t0_space",
"t0_space_iff_exists_is_open_xor_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specialization_order (α : Type*) [topological_space α] [t0_space α] : partial_order α | { .. specialization_preorder α,
.. partial_order.lift (order_dual.to_dual ∘ 𝓝) nhds_injective } | def | specialization_order | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"nhds_injective",
"order_dual.to_dual",
"partial_order.lift",
"specialization_preorder",
"t0_space",
"topological_space"
] | Specialization forms a partial order on a t0 topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minimal_nonempty_closed_subsingleton [t0_space α] {s : set α} (hs : is_closed s)
(hmin : ∀ t ⊆ s, t.nonempty → is_closed t → t = s) :
s.subsingleton | begin
refine λ x hx y hy, of_not_not (λ hxy, _),
rcases exists_is_open_xor_mem hxy with ⟨U, hUo, hU⟩,
wlog h : x ∈ U ∧ y ∉ U,
{ exact this hmin y hy x hx (ne.symm hxy) U hUo hU.symm (hU.resolve_left h), },
cases h with hxU hyU,
have : s \ U = s := hmin (s \ U) (diff_subset _ _) ⟨y, hy, hyU⟩ (hs.sdiff hUo),
... | theorem | minimal_nonempty_closed_subsingleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"exists_is_open_xor_mem",
"is_closed",
"of_not_not",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minimal_nonempty_closed_eq_singleton [t0_space α] {s : set α} (hs : is_closed s)
(hne : s.nonempty) (hmin : ∀ t ⊆ s, t.nonempty → is_closed t → t = s) :
∃ x, s = {x} | exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨hne, minimal_nonempty_closed_subsingleton hs hmin⟩ | theorem | minimal_nonempty_closed_eq_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"minimal_nonempty_closed_subsingleton",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.exists_closed_singleton {α : Type*} [topological_space α]
[t0_space α] [compact_space α] {S : set α} (hS : is_closed S) (hne : S.nonempty) :
∃ (x : α), x ∈ S ∧ is_closed ({x} : set α) | begin
obtain ⟨V, Vsub, Vne, Vcls, hV⟩ := hS.exists_minimal_nonempty_closed_subset hne,
rcases minimal_nonempty_closed_eq_singleton Vcls Vne hV with ⟨x, rfl⟩,
exact ⟨x, Vsub (mem_singleton x), Vcls⟩
end | theorem | is_closed.exists_closed_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"is_closed",
"minimal_nonempty_closed_eq_singleton",
"t0_space",
"topological_space"
] | Given a closed set `S` in a compact T₀ space,
there is some `x ∈ S` such that `{x}` is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minimal_nonempty_open_subsingleton [t0_space α] {s : set α} (hs : is_open s)
(hmin : ∀ t ⊆ s, t.nonempty → is_open t → t = s) :
s.subsingleton | begin
refine λ x hx y hy, of_not_not (λ hxy, _),
rcases exists_is_open_xor_mem hxy with ⟨U, hUo, hU⟩,
wlog h : x ∈ U ∧ y ∉ U,
{ exact this hs hmin y hy x hx (ne.symm hxy) U hUo hU.symm (hU.resolve_left h), },
cases h with hxU hyU,
have : s ∩ U = s := hmin (s ∩ U) (inter_subset_left _ _) ⟨x, hx, hxU⟩ (hs.int... | theorem | minimal_nonempty_open_subsingleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"exists_is_open_xor_mem",
"is_open",
"of_not_not",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minimal_nonempty_open_eq_singleton [t0_space α] {s : set α} (hs : is_open s)
(hne : s.nonempty) (hmin : ∀ t ⊆ s, t.nonempty → is_open t → t = s) :
∃ x, s = {x} | exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨hne, minimal_nonempty_open_subsingleton hs hmin⟩ | theorem | minimal_nonempty_open_eq_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"minimal_nonempty_open_subsingleton",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_open_singleton_of_open_finite [t0_space α] {s : set α} (hfin : s.finite)
(hne : s.nonempty) (ho : is_open s) :
∃ x ∈ s, is_open ({x} : set α) | begin
lift s to finset α using hfin,
induction s using finset.strong_induction_on with s ihs,
rcases em (∃ t ⊂ s, t.nonempty ∧ is_open (t : set α)) with ⟨t, hts, htne, hto⟩|ht,
{ rcases ihs t hts htne hto with ⟨x, hxt, hxo⟩,
exact ⟨x, hts.1 hxt, hxo⟩ },
{ rcases minimal_nonempty_open_eq_singleton ho hne _... | theorem | exists_open_singleton_of_open_finite | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"em",
"finset",
"finset.strong_induction_on",
"is_open",
"lift",
"minimal_nonempty_open_eq_singleton",
"of_not_not",
"t0_space"
] | Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_open_singleton_of_fintype [t0_space α] [finite α] [nonempty α] :
∃ x : α, is_open ({x} : set α) | let ⟨x, _, h⟩ := exists_open_singleton_of_open_finite (set.to_finite _) univ_nonempty
is_open_univ in ⟨x, h⟩ | theorem | exists_open_singleton_of_fintype | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"exists_open_singleton_of_open_finite",
"finite",
"is_open",
"is_open_univ",
"set.to_finite",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space_of_injective_of_continuous [topological_space β] {f : α → β}
(hf : function.injective f) (hf' : continuous f) [t0_space β] : t0_space α | ⟨λ x y h, hf $ (h.map hf').eq⟩ | lemma | t0_space_of_injective_of_continuous | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"t0_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.t0_space [topological_space β] [t0_space β] {f : α → β}
(hf : embedding f) : t0_space α | t0_space_of_injective_of_continuous hf.inj hf.continuous | lemma | embedding.t0_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"embedding",
"t0_space",
"t0_space_of_injective_of_continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.t0_space [t0_space α] {p : α → Prop} : t0_space (subtype p) | embedding_subtype_coe.t0_space | instance | subtype.t0_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space_iff_or_not_mem_closure (α : Type u) [topological_space α] :
t0_space α ↔ (∀ a b : α, a ≠ b → (a ∉ closure ({b} : set α) ∨ b ∉ closure ({a} : set α))) | by simp only [t0_space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_distrib] | theorem | t0_space_iff_or_not_mem_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"inseparable_iff_mem_closure",
"not_and_distrib",
"t0_space",
"t0_space_iff_not_inseparable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space.of_cover (h : ∀ x y, inseparable x y → ∃ s : set α, x ∈ s ∧ y ∈ s ∧ t0_space s) :
t0_space α | begin
refine ⟨λ x y hxy, _⟩,
rcases h x y hxy with ⟨s, hxs, hys, hs⟩, resetI,
lift x to s using hxs, lift y to s using hys,
rw ← subtype_inseparable_iff at hxy,
exact congr_arg coe hxy.eq
end | lemma | t0_space.of_cover | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable",
"lift",
"subtype_inseparable_iff",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space.of_open_cover (h : ∀ x, ∃ s : set α, x ∈ s ∧ is_open s ∧ t0_space s) : t0_space α | t0_space.of_cover $ λ x y hxy,
let ⟨s, hxs, hso, hs⟩ := h x in ⟨s, hxs, (hxy.mem_open_iff hso).1 hxs, hs⟩ | lemma | t0_space.of_open_cover | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"t0_space",
"t0_space.of_cover"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space (α : Type u) [topological_space α] : Prop | (t1 : ∀x, is_closed ({x} : set α)) | class | t1_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"topological_space"
] | A T₁ space, also known as a Fréchet space, is a topological space
where every singleton set is closed. Equivalently, for every pair
`x ≠ y`, there is an open set containing `x` and not `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_singleton [t1_space α] {x : α} : is_closed ({x} : set α) | t1_space.t1 x | lemma | is_closed_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_compl_singleton [t1_space α] {x : α} : is_open ({x}ᶜ : set α) | is_closed_singleton.is_open_compl | lemma | is_open_compl_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_ne [t1_space α] {x : α} : is_open {y | y ≠ x} | is_open_compl_singleton | lemma | is_open_ne | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"is_open_compl_singleton",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.is_open_mul_support [t1_space α] [has_one α] [topological_space β]
{f : β → α} (hf : continuous f) : is_open (mul_support f) | is_open_ne.preimage hf | lemma | continuous.is_open_mul_support | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"is_open",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne.nhds_within_compl_singleton [t1_space α] {x y : α} (h : x ≠ y) :
𝓝[{y}ᶜ] x = 𝓝 x | is_open_ne.nhds_within_eq h | lemma | ne.nhds_within_compl_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne.nhds_within_diff_singleton [t1_space α] {x y : α} (h : x ≠ y) (s : set α) :
𝓝[s \ {y}] x = 𝓝[s] x | begin
rw [diff_eq, inter_comm, nhds_within_inter_of_mem],
exact mem_nhds_within_of_mem_nhds (is_open_ne.mem_nhds h)
end | lemma | ne.nhds_within_diff_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"mem_nhds_within_of_mem_nhds",
"nhds_within_inter_of_mem",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_set_of_eventually_nhds_within [t1_space α] {p : α → Prop} :
is_open {x | ∀ᶠ y in 𝓝[≠] x, p y} | begin
refine is_open_iff_mem_nhds.mpr (λ a ha, _),
filter_upwards [eventually_nhds_nhds_within.mpr ha] with b hb,
by_cases a = b,
{ subst h, exact hb },
{ rw (ne.symm h).nhds_within_compl_singleton at hb,
exact hb.filter_mono nhds_within_le_nhds }
end | lemma | is_open_set_of_eventually_nhds_within | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"nhds_within_le_nhds",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.is_closed [t1_space α] {s : set α} (hs : set.finite s) :
is_closed s | begin
rw ← bUnion_of_singleton s,
exact is_closed_bUnion hs (λ i hi, is_closed_singleton)
end | lemma | set.finite.is_closed | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"is_closed_bUnion",
"is_closed_singleton",
"set.finite",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space.is_topological_basis.exists_mem_of_ne
[t1_space α] {b : set (set α)} (hb : is_topological_basis b) {x y : α} (h : x ≠ y) :
∃ a ∈ b, x ∈ a ∧ y ∉ a | begin
rcases hb.is_open_iff.1 is_open_ne x h with ⟨a, ab, xa, ha⟩,
exact ⟨a, ab, xa, λ h, ha h rfl⟩,
end | lemma | topological_space.is_topological_basis.exists_mem_of_ne | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open_ne",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.coclosed_compact_le_cofinite [t1_space α] :
filter.coclosed_compact α ≤ filter.cofinite | λ s hs, compl_compl s ▸ hs.is_compact.compl_mem_coclosed_compact_of_is_closed hs.is_closed | lemma | filter.coclosed_compact_le_cofinite | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compl_compl",
"filter.coclosed_compact",
"filter.cofinite",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bornology.relatively_compact [t1_space α] : bornology α | { cobounded := filter.coclosed_compact α,
le_cofinite := filter.coclosed_compact_le_cofinite } | def | bornology.relatively_compact | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"bornology",
"filter.coclosed_compact",
"filter.coclosed_compact_le_cofinite",
"t1_space"
] | In a `t1_space`, relatively compact sets form a bornology. Its cobounded filter is
`filter.coclosed_compact`. See also `bornology.in_compact` the bornology of sets contained
in a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bornology.relatively_compact.is_bounded_iff [t1_space α] {s : set α} :
@bornology.is_bounded _ (bornology.relatively_compact α) s ↔ is_compact (closure s) | begin
change sᶜ ∈ filter.coclosed_compact α ↔ _,
rw filter.mem_coclosed_compact,
split,
{ rintros ⟨t, ht₁, ht₂, hst⟩,
rw compl_subset_compl at hst,
exact is_compact_of_is_closed_subset ht₂ is_closed_closure (closure_minimal hst ht₁) },
{ intros h,
exact ⟨closure s, is_closed_closure, h, compl_subs... | lemma | bornology.relatively_compact.is_bounded_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"bornology.is_bounded",
"bornology.relatively_compact",
"closure",
"closure_minimal",
"filter.coclosed_compact",
"filter.mem_coclosed_compact",
"is_closed_closure",
"is_compact",
"is_compact_of_is_closed_subset",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.is_closed [t1_space α] (s : finset α) : is_closed (s : set α) | s.finite_to_set.is_closed | lemma | finset.is_closed | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"finset",
"is_closed",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_tfae (α : Type u) [topological_space α] :
tfae [t1_space α,
∀ x, is_closed ({x} : set α),
∀ x, is_open ({x}ᶜ : set α),
continuous (@cofinite_topology.of α),
∀ ⦃x y : α⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
∀ ⦃x y : α⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : α⦄, x ≠ y → ∃ (U : set α) (hU : is_open U), x... | begin
tfae_have : 1 ↔ 2, from ⟨λ h, h.1, λ h, ⟨h⟩⟩,
tfae_have : 2 ↔ 3, by simp only [is_open_compl_iff],
tfae_have : 5 ↔ 3,
{ refine forall_swap.trans _,
simp only [is_open_iff_mem_nhds, mem_compl_iff, mem_singleton_iff] },
tfae_have : 5 ↔ 6,
by simp only [← subset_compl_singleton_iff, exists_mem_subs... | lemma | t1_space_tfae | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure_subset_iff_is_closed",
"cofinite_topology.is_open_iff'",
"cofinite_topology.of",
"compl_compl",
"continuous",
"continuous_def",
"disjoint",
"disjoint.comm",
"exists_prop",
"is_closed",
"is_open",
"is_open_compl_iff",
"is_open_empty",
"is_open_iff_mem_nhds",
"ne_comm",
"nhds_ba... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_iff_continuous_cofinite_of {α : Type*} [topological_space α] :
t1_space α ↔ continuous (@cofinite_topology.of α) | (t1_space_tfae α).out 0 3 | lemma | t1_space_iff_continuous_cofinite_of | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"cofinite_topology.of",
"continuous",
"t1_space",
"t1_space_tfae",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofinite_topology.continuous_of [t1_space α] : continuous (@cofinite_topology.of α) | t1_space_iff_continuous_cofinite_of.mp ‹_› | lemma | cofinite_topology.continuous_of | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"cofinite_topology.of",
"continuous",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_iff_exists_open : t1_space α ↔
∀ (x y), x ≠ y → (∃ (U : set α) (hU : is_open U), x ∈ U ∧ y ∉ U) | (t1_space_tfae α).out 0 6 | lemma | t1_space_iff_exists_open | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"t1_space",
"t1_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_iff_disjoint_pure_nhds : t1_space α ↔ ∀ ⦃x y : α⦄, x ≠ y → disjoint (pure x) (𝓝 y) | (t1_space_tfae α).out 0 8 | lemma | t1_space_iff_disjoint_pure_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"t1_space",
"t1_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_iff_disjoint_nhds_pure : t1_space α ↔ ∀ ⦃x y : α⦄, x ≠ y → disjoint (𝓝 x) (pure y) | (t1_space_tfae α).out 0 7 | lemma | t1_space_iff_disjoint_nhds_pure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"t1_space",
"t1_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_iff_specializes_imp_eq : t1_space α ↔ ∀ ⦃x y : α⦄, x ⤳ y → x = y | (t1_space_tfae α).out 0 9 | lemma | t1_space_iff_specializes_imp_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space",
"t1_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_pure_nhds [t1_space α] {x y : α} (h : x ≠ y) : disjoint (pure x) (𝓝 y) | t1_space_iff_disjoint_pure_nhds.mp ‹_› h | lemma | disjoint_pure_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nhds_pure [t1_space α] {x y : α} (h : x ≠ y) : disjoint (𝓝 x) (pure y) | t1_space_iff_disjoint_nhds_pure.mp ‹_› h | lemma | disjoint_nhds_pure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.eq [t1_space α] {x y : α} (h : x ⤳ y) : x = y | t1_space_iff_specializes_imp_eq.1 ‹_› h | lemma | specializes.eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_eq [t1_space α] {x y : α} : x ⤳ y ↔ x = y | ⟨specializes.eq, λ h, h ▸ specializes_rfl⟩ | lemma | specializes_iff_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_eq_eq [t1_space α] : (⤳) = @eq α | funext₂ $ λ x y, propext specializes_iff_eq | lemma | specializes_eq_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"funext₂",
"specializes_iff_eq",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_le_nhds_iff [t1_space α] {a b : α} : pure a ≤ 𝓝 b ↔ a = b | specializes_iff_pure.symm.trans specializes_iff_eq | lemma | pure_le_nhds_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"specializes_iff_eq",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_le_nhds_iff [t1_space α] {a b : α} : 𝓝 a ≤ 𝓝 b ↔ a = b | specializes_iff_eq | lemma | nhds_le_nhds_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"specializes_iff_eq",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_antitone {α : Type*} : antitone (@t1_space α) | begin
simp only [antitone, t1_space_iff_continuous_cofinite_of, continuous_iff_le_induced],
exact λ t₁ t₂ h, h.trans
end | lemma | t1_space_antitone | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"antitone",
"continuous_iff_le_induced",
"t1_space",
"t1_space_iff_continuous_cofinite_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_update_of_ne [t1_space α] [decidable_eq α] [topological_space β]
{f : α → β} {s : set α} {x y : α} {z : β} (hne : y ≠ x) :
continuous_within_at (function.update f x z) s y ↔ continuous_within_at f s y | eventually_eq.congr_continuous_within_at
(mem_nhds_within_of_mem_nhds $ mem_of_superset (is_open_ne.mem_nhds hne) $
λ y' hy', function.update_noteq hy' _ _)
(function.update_noteq hne _ _) | lemma | continuous_within_at_update_of_ne | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous_within_at",
"mem_nhds_within_of_mem_nhds",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_update_of_ne [t1_space α] [decidable_eq α] [topological_space β]
{f : α → β} {x y : α} {z : β} (hne : y ≠ x) :
continuous_at (function.update f x z) y ↔ continuous_at f y | by simp only [← continuous_within_at_univ, continuous_within_at_update_of_ne hne] | lemma | continuous_at_update_of_ne | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous_at",
"continuous_within_at_univ",
"continuous_within_at_update_of_ne",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_update_iff [t1_space α] [decidable_eq α] [topological_space β]
{f : α → β} {s : set α} {x : α} {y : β} :
continuous_on (function.update f x y) s ↔
continuous_on f (s \ {x}) ∧ (x ∈ s → tendsto f (𝓝[s \ {x}] x) (𝓝 y)) | begin
rw [continuous_on, ← and_forall_ne x, and_comm],
refine and_congr ⟨λ H z hz, _, λ H z hzx hzs, _⟩ (forall_congr $ λ hxs, _),
{ specialize H z hz.2 hz.1,
rw continuous_within_at_update_of_ne hz.2 at H,
exact H.mono (diff_subset _ _) },
{ rw continuous_within_at_update_of_ne hzx,
refine (H z ⟨hz... | lemma | continuous_on_update_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"and_forall_ne",
"continuous_on",
"continuous_within_at_update_of_ne",
"continuous_within_at_update_same",
"inter_mem_nhds_within",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space_of_injective_of_continuous [topological_space β] {f : α → β}
(hf : function.injective f) (hf' : continuous f) [t1_space β] : t1_space α | t1_space_iff_specializes_imp_eq.2 $ λ x y h, hf (h.map hf').eq | lemma | t1_space_of_injective_of_continuous | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.t1_space [topological_space β] [t1_space β] {f : α → β}
(hf : embedding f) : t1_space α | t1_space_of_injective_of_continuous hf.inj hf.continuous | lemma | embedding.t1_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"embedding",
"t1_space",
"t1_space_of_injective_of_continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.t1_space {α : Type u} [topological_space α] [t1_space α] {p : α → Prop} :
t1_space (subtype p) | embedding_subtype_coe.t1_space | instance | subtype.t1_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space.t0_space [t1_space α] : t0_space α | ⟨λ x y h, h.specializes.eq⟩ | instance | t1_space.t0_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t0_space",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_singleton_mem_nhds_iff [t1_space α] {x y : α} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x | is_open_compl_singleton.mem_nhds_iff | lemma | compl_singleton_mem_nhds_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_singleton_mem_nhds [t1_space α] {x y : α} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y | compl_singleton_mem_nhds_iff.mpr h | lemma | compl_singleton_mem_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_singleton [t1_space α] {a : α} :
closure ({a} : set α) = {a} | is_closed_singleton.closure_eq | lemma | closure_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.closure [t1_space α] {s : set α} (hs : s.subsingleton) :
(closure s).subsingleton | hs.induction_on (by simp) $ λ x, by simp | lemma | set.subsingleton.closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_closure [t1_space α] {s : set α} :
(closure s).subsingleton ↔ s.subsingleton | ⟨λ h, h.anti subset_closure, λ h, h.closure⟩ | lemma | subsingleton_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"subset_closure",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_const {α β} [topological_space α] [topological_space β] [t1_space β] {y : β} :
is_closed_map (function.const α y) | is_closed_map.of_nonempty $ λ s hs h2s, by simp_rw [h2s.image_const, is_closed_singleton] | lemma | is_closed_map_const | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed_map",
"is_closed_map.of_nonempty",
"is_closed_singleton",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_insert_of_ne [t1_space α] {x y : α} {s : set α} (hxy : x ≠ y) :
𝓝[insert y s] x = 𝓝[s] x | begin
refine le_antisymm (λ t ht, _) (nhds_within_mono x $ subset_insert y s),
obtain ⟨o, ho, hxo, host⟩ := mem_nhds_within.mp ht,
refine mem_nhds_within.mpr ⟨o \ {y}, ho.sdiff is_closed_singleton, ⟨hxo, hxy⟩, _⟩,
rw [inter_insert_of_not_mem $ not_mem_diff_of_mem (mem_singleton y)],
exact (inter_subset_inter ... | lemma | nhds_within_insert_of_ne | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed_singleton",
"nhds_within_mono",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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